By Proposition REF , it suffices to show that
\(\Vert \chi _{\nu }^{k}(D)e^{it\phi (D)}f\Vert _{L^{p}(\mathbb {R}^{d})}\lesssim (1+|t|)^{2s(p)}\Vert \chi _{\nu }^{k}(D)f\Vert _{L^{p}(\mathbb {R}^{d})}\) 
for all \(t\in \mathbb {R}\) , \(f\in \mathcal {B}^{s}_{p,q,q}(\mathbb {R}^{d})\) , \(k\in \mathbb {N}_{0}\)  and \(\nu \in \Theta _{k}\) . This is clearly true for \(p=2\) . Hence, by interpolation and duality, it suffices to show the statement for \(p=1\) .
To do so, we rely on a dilation argument. Let \(\lbrace \tilde{\chi }_{\omega }^{m}\mid m\in \mathbb {N}_{0},\omega \in \Theta _{m}\rbrace \subseteq C^{\infty }_{c}(\mathbb {R}^{d})\)  be a collection of cut-offs with similar support and decay properties as the \(\chi _{\omega }^{m}\) , but such that \(\tilde{\chi }_{\omega }^{m}\equiv 1\)  on \(\mathrm {supp}(\chi _{\omega }^{m})\) . Then
\(\sup _{|t|\le 4}\Vert \tilde{\chi }_{m}^{\omega }(D)e^{it\phi (D)}g\Vert _{L^{1}(\mathbb {R}^{d})}\lesssim \Vert g\Vert _{L^{1}(\mathbb {R}^{d})}\) 
for all \(g\in L^{1}(\mathbb {R}^{d})\) , as follows either from kernel bounds (see [1]}), or from the boundedness of \(e^{it\phi (D)}\)  on \(\mathcal {H}^{1}_{FIO}(\mathbb {R}^{d})\) . Either way, we may thus suppose that \(|t|> 4\) .
Let \(l\ge 2\)  be such that \(2^{l}<|t|\le 2^{l+1}\) . Then the dilated function \(\chi _{\nu }^{k}(\frac{\cdot }{|t|})\)  satisfies
\(\mathrm {supp}(\chi _{\nu }^{k}(\tfrac{\cdot }{|t|}))\subseteq \lbrace \xi \in \mathbb {R}^{d}\mid 2^{k+l-1}\le |\xi |\le 2^{k+l+2},|\hat{\xi }-\nu |\le 2^{1-k/2}\rbrace .\) 
Hence one has
\(\chi ^{k}_{\nu }(\tfrac{\cdot }{|t|})=\sum _{m=k+l-2}^{k+l+3}\sum _{\omega \in \tilde{\Theta }_{m}}\tilde{\chi }^{m}_{\omega }\chi ^{k}_{\nu }(\tfrac{\cdot }{|t|}),\) 
where each \(\tilde{\Theta }_{m}\)  has approximately \(2^{l(n-1)/2}\eqsim |t|^{(n-1)/2}\)  elements. Write \(f_{t}(y):=|t|^{d}f(|t|y)\)  for \(y\in \mathbb {R}^{d}\) . Then it suffices to combine (REF ) with dilation arguments:
\(\Vert \chi _{\nu }^{k}(D)e^{it\phi (D)}f\Vert _{L^{1}(\mathbb {R}^{d})}&=|t|^{-d}\big \Vert \big (\chi _{\nu }^{k}(\tfrac{D}{|t|})e^{i\phi (D)}f_{t}\big )\big (\tfrac{\cdot }{|t|}\big )\big \Vert _{L^{1}(\mathbb {R}^{d})}\\&=\Vert \chi _{\nu }^{k}(\tfrac{D}{|t|})e^{i\phi (D)}f_{t}\Vert _{L^{1}(\mathbb {R}^{d})}\\&\le \sum _{m=k+l-2}^{k+l+3}\sum _{\omega \in \tilde{\Theta }_{m}}\big \Vert \tilde{\chi }^{m}_{\omega }(D)e^{i\phi (D)}\chi ^{k}_{\nu }(\tfrac{D}{|t|})f_{t}\big \Vert _{L^{1}(\mathbb {R}^{d})}\\&\lesssim |t|^{(n-1)/2}\Vert \chi ^{k}_{\nu }(\tfrac{D}{|t|})f_{t}\Vert _{L^{1}(\mathbb {R}^{d})}=|t|^{(n-1)/2}\Vert \chi ^{k}_{\nu }(D)f)\Vert _{L^{1}(\mathbb {R}^{d})},\) 

where we used in particular the homogeneity of \(\phi \) .
